how to express Chebyshev sequence constants for function approximations in an interval other than [-1,1]?

Question

To approximate the function with Cheboshev polynomials, it is necessary to operate on the interval [-1,1]. How can these constants be recalculated if I want to approximate on another interval?

specifically, I use maple and the following loop:

(https://i.stack.imgur.com/TWT74.png)

but I don't know how to modify the function to calculate in an interval, for example [-pi,pi]

Answer 1

If you have a function f(x) defined on [-pi, pi] then you can transform it to a function g(u) on [-1, 1] by a linear change of variable:

u = -1 + 2 * (x + pi) / (2*pi).

Then you can approximate g by a polynomial P(u), and then transform P(u) to the polynomial Q(x) by the inverse change of variables:

x = -pi + (2*pi) * (u + 1) / 2.